Understanding Momentum in Continuous Mass Flow Problems

[unscientific]

Homework Statement

From 2.2 Worked Examplehttp://ocw.mit.edu/courses/physics/8-01sc-physics-i-classical-mechanics-fall-2010/continuous-mass-flow/MIT8_01SC_coursenotes19.pdf"

Emptying a Freight Car - A freight car of mass m_c
contains a mass of sand m_s
At t = 0 a constant horizontal force of magnitude F is applied in the direction of rolling and at the same time a port in the bottom is opened to let the sand flow out at the constant rate b = dm_s/ dt . Find the speed of the freight car when all the sand is gone. Assume that the freight car is at rest at t = 0 .

Homework Equations

It is written that the momentum of vehicle at time t is ( $$\Delta$$m + m ) v

The Attempt at a Solution

Shouldn't the momentum of the vehicle at time t be just (m_c(t))v ? by adding $$\Delta$$m you're doing it twice!

I don't understand how they got the momentum for time t+ $$\Delta$$t too! shouldn't it be (m_c - $$\Delta$$m)(v + $$\Delta$$v)?

 

[matematikawan]

Homework Statement

The Attempt at a Solution

Shouldn't the momentum of the vehicle at time t be just (m_c(t))v ? by adding $$\Delta$$m you're doing it twice!

I don't understand how they got the momentum for time t+ $$\Delta$$t too! shouldn't it be (m_c - $$\Delta$$m)(v + $$\Delta$$v)?

I would agree with you that the momentum at time t is (m_c(t))v. But disagree with the momentum at t+ $$\Delta$$t. It should be
$$P(t+\delta t)=(m-\delta m)(v+\delta v)+\delta mv.$$

So the final result should remain as $$F=m\frac{dv}{dt}.$$